A motorcycle is depreciating at $17 \%$ per year, every year. A student's $\$ 13,100$ motorcycle depreciating at this rate can be modelled by the equation $V(t)=13,100(0.83)^t$. What is an equivalent equation for this vehicle as a monthly depreciation rate?
A. $13,100(0.8300)^{12 t}$
B. $13,100(0.9846)^{12 t}$
C. $13,100(1.17)^{-t}$
D. $13,100(0.9846)^t$
Answer (2)
The problem provides an annual depreciation model V ( t ) = 13100 ( 0.83 ) t and asks for an equivalent monthly model.
We express the monthly model as V ( t ) = 13100 ( x ) 12 t , where x is the monthly depreciation factor.
We find x by solving ( 0.83 ) = ( x ) 12 , which gives x = ( 0.83 ) 12 1 ≈ 0.9846 .
The equivalent equation is therefore 13 , 100 ( 0.9846 ) 12 t .
Explanation
Understanding the Problem We are given the equation V ( t ) = 13100 ( 0.83 ) t which models the depreciation of a motorcycle at a rate of 17% per year. We want to find an equivalent equation that models the depreciation on a monthly basis.
Setting up the Equivalent Equation Let the monthly depreciation rate be r m . Then the equivalent equation will be of the form V ( t ) = 13100 ( 1 − r m ) 12 t . We want to find the value of 1 − r m such that ( 0.83 ) t = ( 1 − r m ) 12 t .
Finding the Monthly Depreciation Factor Taking the t -th root of both sides, we have 0.83 = ( 1 − r m ) 12 . Now, we take the 12th root of both sides: ( 0.83 ) 12 1 = 1 − r m .
Calculating the 12th Root We calculate ( 0.83 ) 12 1 : ( 0.83 ) 12 1 ≈ 0.9846 . Therefore, the equivalent equation is V ( t ) = 13100 ( 0.9846 ) 12 t .
Final Answer The equivalent equation for the motorcycle's value with a monthly depreciation rate is V ( t ) = 13100 ( 0.9846 ) 12 t .
Examples
Understanding depreciation is crucial in personal finance. For instance, when buying a car, knowing the depreciation rate helps you estimate its future value. If a car costs 25 , 000 an dd e p rec ia t es a t 15 V(t) = 25000(0.85)^t t o p re d i c t i t s w or t ha f t er t$ years. This knowledge aids in making informed decisions about when to sell or trade in the vehicle, optimizing your financial planning.
The equivalent equation for the motorcycle's value with a monthly depreciation rate is given by V ( t ) = 13100 ( 0.9846 ) 12 t . The monthly depreciation factor was derived from the annual depreciation rate. Therefore, the correct option is B .
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